Cyclic cohomology
Cyclic cohomology was developed as a replacement of the de Rham theory in the context of non-commutative algebras. It was discovered independently by B. Feigin and B. Tsygan as a non-commutative analogue of de Rham cohomology and by A. Connes as the cohomological structure involved in the computation of indices of elliptic operators (cf. Index formulas) and the range of a Chern character defined on -homology (cf. also
-theory). It plays a fundamental role in various generalizations of index theorems to foliations and coverings (higher indices of Connes and H. Moscovici). Among the more classical applications there are the proof of the Novikov conjecture for a large class of groups and local formulas for Pontryagin classes.
Suppose that is a unital algebra over a field
of characteristic zero. Let
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The complex computes the Hochschild cohomology
of
with values in the dual
-bimodule
. The subspace of cyclic cochains,
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is closed under the Hochschild coboundary mapping and the cohomology of the induced complex
is called the cyclic cohomology of the algebra
, denoted by
. This is a contravariant functor from the category of associative algebras to the category of linear spaces. The inclusion of complexes
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gives a long exact sequence of cohomology which, in this case, has the form
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and is usually called the Connes–Gysin exact sequence. The induced spectral sequence is one of the main tools for the computation of cyclic cohomology. The periodicity operator gives rise to the stabilized version (
-graded) of cyclic cohomology given by the direct limit
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and called periodic cyclic cohomology. Both cyclic and periodic cyclic cohomologies admit external products:
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Some examples are:
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Any tracial functional (i.e.
for all
) on the algebra
defines an element of
.
Let be a smooth, closed manifold and
. Restricting to the continuous (in the
-topology) cochains, one finds
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where is the space of de Rham
-currents on
and the operator
coincides with the transpose of the de Rham differential
. In particular, the periodic cyclic cohomology of
coincides with de Rham homology of
. An explicit mapping is given by
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The definitions given above can be easily extended to non-unital algebras.
Properties of periodic cyclic cohomology.
Stability. The mapping
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is an isomorphism. This holds also for (non-periodic)) cyclic cohomology. Here, is the mapping induced by
from the algebra of
-matrices over
to
.
Excision. Given an exact sequence of algebras
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there exists an associated six-term exact sequence
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The excision holds also in the case of Fréchet algebras (cf. also Fréchet algebra) if the quotient mapping has a continuous (linear) lifting. For cyclic cohomology, under a certain condition on the ideal
(called
-unitality) there exists a corresponding long exact sequence.
Homotopy invariance. Suppose that, in the context of topological algebras and continuous cochains, a differentiable family of homomorphisms is given:
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The corresponding mappings on the periodic cyclic cohomology are
-independent. In the context of cyclic cohomology the corresponding result says that inner derivations act trivially.
Pairing with -theory. Suppose that
is a Banach algebra and that
is a dense subalgebra closed under holomorphic functional calculus in
. Suppose that
has been given a Fréchet topology under which its imbedding into
is continuous. Restricting to continuous cochains, there exists a pairing
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On the level of idempotents , this pairing is given by
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This pairing is consistent with the six-term exact sequences in periodic cyclic cohomology and -theory.
Bott periodicity. In the topological context, let be the projective tensor product of
with the algebra of rapidly decreasing functions on
. Let
denote the cyclic cocycle on
given by the current
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The mapping
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is an isomorphism.
References
[a1] | A. Connes, "Noncommutative geometry" , Acad. Press (1994) |
[a2] | J. Cuntz, D. Quillen, "Algebra extensions and nonsingularity" J. Amer. Math. Soc. , 8 , Amer. Math. Soc. (1995) pp. 251–289 |
[a3] | J-L. Loday, "Cyclic homology" , Springer (1991) |
Cyclic cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_cohomology&oldid=11855